Greece: The Beginnings to Euclid - Mathematics Educationmathed.byu.edu/~williams/Classes/300F2011/PDFs/PPTs/Greece 1.pdf · particularly geometry, to Greece, from Egypt. ... of Geometry

Greece: The Beginnings to Euclid

Chapters 2 and 3

Timeline and Overview

MycenaeanMinoan GREECEChristianRomanHellenisticClassicalArchaicDark

500 AD0 AD500 BC3000 BC 2500 BC 2000 BC 1500 BC 1000 BC

MESOPOTAM IA

EGYPTInt

Int

1000 BC1500 BC2000 BC2500 BC3000 BC

New KingdomMiddle KingdomIntOld KingdomArchaic

AssyriaOld BabylonAkkadiaSumaria

Minoans were traders. We have not deciphered their written language, Linear A.Mycenaeans were warriors. They fought against Troy in Homer’s epic.The Greek Dark Ages – depopulation, famine, loss of writing. Archaic – New alphabet from Phoenecia, added vowel sounds, became Greek alphabet.This is where we pick up the mathematical story of Greece.

Archaic Smiles:

Sources

• How do we know what we know about Greek mathematics?

• Surprisingly, we have even fewer original manuscripts from ancient Greece than from Egypt or Mesopotamia. So our knowledge may be less reliable, coming as it does from secondary sources:– Byzantine Greek codices (manuscript books) written 500‐1500 years after the Greek works were composed.

– Arabic translations of Greek works and Latin translations of the Arabic versions.

CHAPTER 2

Numeration System

• What did they use for numerals?• How did they write fractions?• What do we know about their calculation?

Greek Number System

• Each unit (1, 2, …, 9) was assigned a separate letter, each tens (10, 20, …, 90) a separate letter, and each hundreds (100, 200, …, 900) a separate letter. This requires 27 letters, so 3 obsolete characters were added.

• A ‘ was used after a letter to indicate a numeral, and a , was used before a letter to multiply its value by 1000.

Greek Numerals

Letter Value Letter Value Letter Value

α′ 1 ι′ 10 ρ′ 100

β′ 2 κ′ 20 σ′ 200

γ′ 3 λ′ 30 τ′ 300

δ′ 4 μ′ 40 υ′ 400

ε′ 5 ν′ 50 φ′ 500

ϝ′ or ϛ′ or στ′ 6 ξ′ 60 χ′ 600

ζ′ 7 ο′ 70 ψ′ 700

η′ 8 π′ 80 ω′ 800

θ′ 9 ϟ′ 90 ϡ′ 900

Thales of Miletus

• What was he remembered for?

Thales of Miletus• Thales of Miletus (about 600 BC) learned his math in Egypt

and Mesopotamia. He is sometimes referred to as the “Father of Demonstrative Mathematics,” because he not only stated mathematical facts and procedures, but proved them. This is in contrast to what we know about Egypt, Mesopotamia, and other older cultures, and is why Ancient Greece figures so prominently in the history of mathematics.

• May have introduced the mathematical sciences, particularly geometry, to Greece, from Egypt.

• One of the Seven Wise Men, the only one “whose wisdom stepped, in speculation, beyond the limits of practical utility” (Plutarch).

Thales of Miletus

• “Tradition” holds that Thales first proved:– Vertical angles are equal– The base angles of an isosceles triangle are equal– An angle inscribed in a semicircle is a right angle– A circle is bisected by any diameter– Two triangles are congruent if they have two angles and the included side equal.

Thales of Miletus

• There are stories about Thales predicting an eclipse. However, it wasn’t uncommon to attribute such things to men with reputations for wisdom.

• Aristotle relates a story of how Thales used his skills to deduce that the next season's olive crop would be a very large one. He therefore bought all the olive presses and then was able to make a fortune when the bumper olive crop did indeed arrive.

• No original writings remain.

Pythagoras and His Followers

• Who were they?• In what sense did they believe that “number was the substance of all things?”

In the sense that• “Numbers formed the basic organizing principle of the universe”

Pythagoras

• By “tradition,” Pythagoras (585 – 501 BC) also studied in Egypt, perhaps at the urging of Thales himself. From Egypt, he was taken prisoner by the Persians. Thus he too probably learned much from Egypt and Babylonian traditions.

• Most importantly, he founded the Pythagorean Brotherhood (although women were equal members), a society of scholars with cult‐like behaviors and beliefs.

• Essentially, it was a commune. Buncha dang hippies.

Pythagoreans

• Never ate meat or beans (beans were sacred), • Never hunted and used no wool, • Dressed in white and drank no wine, • Did not pick up anything that had fallen, • Did not stir a fire with an iron, • Did not look in mirrors beside a light, and • Slept in white linen bedding.

Pythagoreans

• Their symbol was pentagram, and they used various rituals to strengthen solidarity.

• They engaged in frequent exercise, silent contemplation, and study of mathematics.

• They believed in transmigration of souls. • They also became politically strong, in their local area, leading to the burning of their meeting house about 501 BC.

Pythagoreans

• They believed property was communal, and so even their mathematical discoveries were communal. Thus anything any Pythagorean discovered became attributed to Pythagoras. It is therefore difficult to know what hediscovered, and what was discovered by his community of followers.

Pythagorean Mathematics

• Pythagoras saw the study of mathematics as a purifier of the soul, just like he considered music as purifying.

• He had long played the seven string lyre, and learned how harmonious the vibrating strings sounded when the lengths of the strings were in ratios of whole numbers, such as 2:1, 3:2, 4:3.


• They tended to work with ratios rather than direct measurements. Thus they would say that the areas of two circles had the same ratio as the squares produced from their two radii.

• This is how they would express the idea that the area of a circle was proportional to the square of its radius.

Incommensurability

• What is meant by numbers being “incommensurable?”

• What about two magnitudes?


• Two segments are commensurable if there is a smaller segment that divides evenly into both. (To us, the ratio of commensurable lengths is a rational number.) The Pythagorean Hippasus* discovered that the diagonal of a square is not commensurable with its side, contrary to the Pythagorean belief that any two segments were commensurable. Legend has it that they tossed him from a boat.*

* Your author doesn’t believe this, though.


• What we would today interpret as irrational numbers – ratios of incommensurable segments –may have caused some distress, and in fact may have contributed to the Greek preference for line segments, instead of their lengths, that was prevalent throughout the 1000 years of Greek mathematics. It is true that Greek mathematics tended to be very geometrical rather than arithmetic or (what we would call) algebraic.

Hippocrates of Chios

• Why is Hippocrates of Chios remembered?


• Hippocrates of Chios (460 – 380 BC) wrote a book of “Elements” that we have no copy of. He also performed the quadrature of a particular lune. Quadrature in general is the art of finding a square with the same area as a given (less regular) figure. Notice that the problem is not to find numbers, but to construct, with compass and straightedge, an actual square. Hippocrates was able to “square,” or perform the quadrature of, 3 different lunes.


• The next person to have any success in this was Euler, about 2000 years later, who performed the quadrature of 2 more lunes(although some claim those two were done by Martin Johan Wallenius in 1766 as well).

• Finally, it was proved in 1994 by Tschebatorewand Dorodnow that the rest cannot be squared.

The Five Squarable Lunes

Speaking of Quadrature. . .

• We should mention the three classic problems of (Greek) antiquity:– The quadrature of the circle; that is, to find the square of an area equal to that of a given circle

– The trisection of an angle; that is, the problem of dividing a given angle into three equal parts

– The duplication of a cube; that is, to find the side of a cube of which the volume is twice that of a given cube

*All with straightedge and compass*

The Three Classic Problems

• What role did these play in early Greek mathematics?

• “… a central goal of Greek mathematics was geometrical problem solving, and … to a large extent, the great body of theorems found in the major extant works of Greek mathematics served as logical underpinnings for these solutions.” ‐‐ Katz

The Three Classic Problems• The duplication problem was proved impossible first by Descartes in 1637.

• The trisection problem was put to rest by Pierre Laurent Wantzel, who published his proofs in a paper called "Research on the Means of Knowing If a Problem of Geometry Can Be Solved with Compass and Straight‐edge", in 1837.

• In 1882, the German mathematician Ferdinand Lindemann proved that the quadrature of the circle is impossible, by proving that is transcendental – not the root of any polynomial equation with integer coefficients.

Plato

• Do you understand Plato’s distinction between the ideal or perfect, and the physical?

• We’ll come back to Plato later in the course.

Aristotle

• Aristotle’s contributions to math…….– Logic– Numbers versus magnitudes– Answers to Zeno

CHAPTER 3

The Library at Alexandria

• What do we know about it?

• Well, a little history…

Alexander the Great

• In about 352 BC, the Macedonian King Philip II began to unify the numerous Greek city‐states into one kingdom. After his death, his son, Alexander (the Great) continued the conquest of everything between Macedonia and India.

Alexander the Great’s Empire

Alexandria

• Alexander planned and built the city of Alexandria in Egypt, on the west end of the Nile river delta. Although Alexander died before the city was complete, it remained the capital of Egypt for nearly a thousand years. Ptolemy, one of Alexander’s generals, took over the Egyptian part of his empire. A son, Ptolemy II, built a library and museum, and Ptolemy III populated it with books (scrolls, really).

Euclid

• The Library of Alexandria, founded about 300 BC, became the center of learning. It had reading rooms, lecture rooms, meeting rooms, a dining hall, and gardens to walk in (sounds like a modern‐day college).

• It is to this library that Euclid came to study and teach.

Euclid’s Elements

• What can we remember about the source of our current version of the Elements?

• Original (lost)• Commentary and copy by Theon of Alexandria (300‐400 CE)– Copy at Oxford (888 CE)

• 900 CE, copy from version earlier than Theon’s, in Vatican library

• 1880’s, comparison by J. L. Heiberg• 1908, Thomas Heath, pretty much the standard today.

Euclid’s Elements

• What can we remember about the content of the Elements?

• Geometry, plane and solid• Ratios of numbers and of magnitudes• Geometric algebra• Number theory• Polyhedra

Euclid’s Elements

• Book I: Basic plane geometry• Book II: “Geometric” algebra• Book III: Circles • Book IV: Inscribing and circumscribing figures• Book V: Extending Eudoxus’ ideas of ratio• Book VI: Similarity of figures• Books VII – IX: Number theory• Book X: Incommensurable magnitudes• Books XI – XIII: Solid (3 dimensional) geometry

Euclid’s Elements

• 13 books, 465 propositions in plane and solid geometry and number theory.

• Few if any results are original to Euclid; it is likely a compendium of already‐known results. In fact it has been suggested that the first books of Euclid’s Elementsmay be taken from the Elements of Hippocrates of Chios.

Euclid’s Elements

• What is important is the logical structure of the books.

• He gave us an axiomatic development of geometry:– 23 definitions– 5 postulates– 5 common notions

Euclid’s Elements

• Most of each book consisted of propositions which were proved using only the definitions, common notions, and postulates, as well as any propositions previously proved. Thus Proposition I.3 may be proved using only the common notions, postulates, definitions, and Propositions I.1 and I.2.

Euclid’s Elements

• Some of Euclid’s definitions:– A point is that which has no part (1).– A line is breadthless length (2).– A straight line is a line which lies evenly with the points on itself (4).

– When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands (10).

– A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another (15).

Euclid’s ElementsEuclid’s five postulates:

• To draw a straight line from any point to any point• To produce a finite straight line continuously in a straight

line.• To describe a circle with any center and distance.• That all right angles are equal to one another.• That, if a straight line falling on two straight lines makes the

interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Euclid’s Elements

Euclid’s Common Notions:• Things which are equal the same thing are also equal to one another

• If equals be added to equals, the wholes are equal

• If equals be subtracted from equals, the remainders are equal

• Things which coincide with one another are equal to one another

• The whole is greater than the part.

Euclid’s ElementsProposition 1: On a given finite straight line to construct an equilateral

triangle.• Let AB be the given finite straight line. It is required to construct an

equilateral triangle on the straight line AB. Describe the circle BCDwith center A and radius AB (Postulate 3). Again describe the circle ACE with center B and radius BA (Postulate 3). Join the straight lines CA and CB from the point C at which the circles cut one another to the points A and B (Postulate 1).

C

A BD

E

Euclid’s Elements

• Now, since the point A is the center of the circle CDB, therefore ACequals AB (Definition 15). Again, since the point B is the center of the circle CAE, therefore BC equals BA (Definition 15). But AC was proved equal to AB, therefore each of the straight lines AC and BCequals AB. And things which equal the same thing also equal one another, therefore AC also equals BC (Common Notion 1). Therefore the three straight lines AC, AB, and BC equal one another. Therefore the triangle ABC is equilateral, and it has been constructed on the given finite straight line AB. Being what it was required to do.

C

A BD

E

Euclid’s Elements

• Some problems:– Definitions: The first few are vague and intuitive (that which has no part? breadthless length?). Later definitions sometimes leave out parts (what does it mean for an angle to be “greater” than another?) Many he never uses or even refers to later in the book.

– The fix: some terms are defined, while some remain undefined. Gaps are filled in definitions.

Euclid’s Elements

• More Problems– Proofs: Use unstated assumptions. For example, in the proof of Proposition 1, how do we know that the point C exists at the intersection of two circles? What does it mean for one point to be “between” two other points on a line? Diagrams can make these ideas clear and convincing, but it does not meet modern standards of rigor.

– The fix: make all unstated assumptions explicit and prove them first if possible. Add postulates if necessary.

Aside: Fixing Euclid’s Elements• In the late 1800's David Hilbert developed a set of 20 axioms that

made explicit all the assumptions needed to complete Euclid’s program. Like Euclid’s axioms, Hilbert’s were synthetic, that is, they did not depend on properties of the real numbers made explicit though coordinates (as in Descartes’ analytic geometry).

• In 1932, Birkhoff developed a system of axioms that made explicit use of real numbers (e.g. made lines essentially equivalent to real‐number lines).

• In the 1960's the School Mathematics Study Group (SMSG) developed set of axioms somewhat like Birkhoff’s, but especially suited to high school study. They differed in that they were not a “minimal” list of axioms but included some axioms that could be proved from the others in order to avoid more tedious technical proofs.

Other Axiom Systems for GeometryDeveloper Axiom Set Comments

Hilbert 20 Axioms “Fixed” EuclidDoes not explicitly depend on Real Numbers, Set Theory, etc.

Many AxiomsDetailed Technical Proofs of the ObviousImplicitly depends of the real numbers and Set Theory

Birkhoff Only 4 AxiomsIncorporates Real Numbers and MeasurementMore accessible

Only Euclidean (fourth axiom is about similar triangles)

MacLane 14 Axioms including “continuity axiom”

SMSG 22 Axioms Designed for High School Geometry

Some unnecessary axioms

UCSMP Lots of Axioms RedundancyUses Reflection

Venema(a current text)

12 Axioms Explicitly uses real numbers and set theory. Probably closest to Birkhoff’s.

David Hilbert

• 1862 (Königsberg) – 1943 (Göttngen)

• Grundlagen der Geometrie (1899)

• 23 Unsolved Problems (1900)

• Modern Axiomatics and Proof Theory

• Wir müssen wissen. Wir werden wissen.

Is it just me, or do these two look alike?

OK, Back to the Elements

• As an overview of Book I, let’s look at how theorems build to the proof of the Pythagorean Theorem in Proposition 47.

• Prop. I.4 – If two triangles have two sides of one triangle equal to two sides of

the other triangle plus the angle between the sides that are equal in each triangle is the same, then the two triangles are congruent

Moving Toward I.47 ‐ I.4

Moving Toward I.47 – I.14

• Prop. I.14– Two adjacent right angles make a straight line.

– Definition 10 asserted the converse, that a perpendicular erected on a straight line makes two right angles.

• Prop. I.41– The area of a triangle is one half the area of a parallelogram with the same base and height.



• Prop. I.31– Given a line and a point not on the line, a line through the point can be constructed parallel to the first line.

• Prop. I.46– Given a straight line, a square can be constructed with the line as one side.


And Now: Proposition I.47

• In right‐angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.

Proof of Proposition I.47

• Draw a line parallel to the sides of the largest square, from the right angle vertex, A, to the far side of the triangle subtending it, L.

• Connect the points FC and AD, making ∆FBC and ΔABD

• The two shaded triangles are congruent (by Prop. I.4) because the shorter sides are respectively sides of the constructed squares and the angle between them is an angle of the original right triangle, plus a right angle from a square.


• The shaded triangle has the same base (BD) as the shaded rectangle, and the same height (DL), so it has exactly half the area of the rectangle, by Proposition I.41.


• Similarly, the other shaded triangle has half the area of the small square since it has the same base (FB) and height (GF). (Here is where he needed that G, A and C were all on one line.


• Since the triangles had equal areas, twice their areas must also be equal to each other (Common notion 2), hence the shaded square and rectangle must also be equal to each other.


• By the same reasoning, triangles constructed around the other non‐right vertex of the original triangle can also be shown to be congruent.


• And similarly, the other square and rectangle are also equal in area.


• And finally, since the square across from the right angle consists of the two rectangles which have been shown equal to the squares on the sides of the right triangle, those squares together are equal in area to the square across from the right angle.



Just for fun, James Garfield’s proof:

• Start with a right triangle with legs of length b and a, and hypotenuse of length c. Extend the side of length b to length a + b, and construct a perpendicular of length b. Form segments as shown.

a

cb

b

a c

Garfield’s Proof

• Now the area of the trapezoid formed can be calculated in two different ways: as a trapezoid, the area is ½ the sum of the bases, times the height:

a

cb

b

a c

Garfield’s Proof

• On the other hand, we can calculate the area of the half square of length c and add it to the sum of the two triangles:

•

a

cb

b

a c

Garfield’s Proof

So,

So, ya think Bush or Obama could do that?

a

cb

b

a c

Book II: Geometric Algebra

• An example of geometrical algebra is finding the square root of a magnitude by geometric construction:

• Let AB = a, and extend AB to C with BC = 1.

• , so .

1a

x

D

MA

CB

Proposition II‐5

• If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half.

• Huh?

Proposition II‐5

• If a straight line

Proposition II‐5

• If a straight line is cut into equal and unequal segments,

a xy

MA BC

Proposition II‐5

• If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole

x

y

x

a xy

MA BC

Proposition II‐5

• If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section

a

a

x

y

x

a xy

MA BC

Proposition II‐5

• If a straight line is cut into equal and unequal segments, then the rectangle contained by the unequal segments of the whole together with the square on the straight line between the points of section equals the square on the half.

x

xa

a

x

y

x

a xy

MA BC

Proposition II – 5 – Why?

• Well, since the length of segment AM is the same as x+a, this is an x by x+a rectangle.

x

xa

a

x

y

x

a xy

MA BC

Proposition II – 5 – Why?

• This is also an x by x+a rectangle.

x

xa

a

x

y

x

a xy

MA BC

Proposition II – 5 – Why?• So the rectangle on the two unequal segments, the left hand purple together with the brown rectangle, added to the small grey square, will add up to the square on the half (also purple together with brown and grey).

x

xa

a

x

y

x

a xy

MA BC

OK, Where’s the Algebra?

• Suppose we have the two equations

which are typical of Babylonian problems that gave rise to what we would call quadratic equations.


In the diagram, D is the area of the orange rectangle, and b is the length of the line segment AB.

a

a xy

MA BC


Also, and . Then II‐5

says that , or .

a

a xy

MA BC


Now, we just need to construct the value asomehow. This can be done with ruler and compass.

Finding :

M

1D

x

Finding a:

22

Finally,

• Recall that and . It is now easy to construct segments of these two lengths.

• One notion, discussed extensively in Book V, was Eudoxus’ notion. This was applied to magnitudes in Book VI where Euclid deals with similar figures.

• For both magnitudes and numbers, there was a different notion, as discussed in Books VII and X. This used Euclidean algorithm, and is sometimes called anthyphairesis.

Ratios and Proportion in Euclid

Eudoxus of Cnidos

• Eudoxus of Cnidos (about 370 BC) studied under the Pythagorean Archytas of Tarentum, then under Plato for a time, and went to Egypt to study astronomy.

• Developed a theory based on ratios of magnitudes.

• Eudoxus was also responsible for advancing the method of exhaustion, reminiscent of ideas from our modern calculus, which was later used by Archimedes to determine the area of a circle.

Eudoxus of Cnidos

• Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.

Eudoxus and Ratios

• In our notation: If we take four quantities: a, b, c, and d, to say that a / b = c / d we do the following:

• For any two arbitrary integers m and n, if it happens that m∙a > n∙b, then we must also have m∙c > n∙d. If it happens that m∙a = n∙b, then we must also have m∙c = n∙d. Finally, if it happens that m∙a < n∙b, then we must also have m∙c < n∙d.

Eudoxus and Ratios

• In other words we say that a / b = c / d if for any two arbitrary integers m and n, and both have exactly the same order relationships to .

• Or again, two numbers are equal if they stand in exactly the same order relationships to every rational number.

Eudoxus and Ratios

• This avoids any problems with commensurability. It essentially defines numbers in terms of their relationships to rational numbers. Equivalent to a “Dedekind cut.”

• The Eudoxian definition of proportionality uses the quantifier, "for every ..." to address the infinite and the infinitely small, just as Cauchy and Weierstrass would do centuries later, giving rise to our epsilon‐delta definitions of limit and continuity.

Eudoxus and Ratio

• When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science for a century, until it was replaced by the algebraic methods of Descartes.

The Other Ratio: Anthyphairesis

• Working with quantities (say the lengths A and B of two line segments). We try to measure A with copies of B, and find that there is a remainder of R1 < B.

• We then try to measure B by copies of R1. If we are successful, then R1 will measure both A and B. If not, we find there is a remainder of R2 < R1.

Euclidean Algorithm

• We then continue by attempting to measure R1 by R2, and so on. If the process terminates by finding a quantity that measures both original quantities, we know those quantities are commensurable.

• For quantities such as lengths, we know this process may not terminate, as when we begin with the diagonal and side of a square.

• If we begin with whole numbers, we know it must terminate because the sequence of remainders is a decreasing sequence of positive integers, which must terminate (in the worse case, it would terminate with 1 being the GCD).

Euclidean Algorithm

• Example with 67 and 21

Example

67 3 21 421 5 4 14 4 1 0

• Let’s look at 67x5 and 21x5, or 335 and 105:

Example

335 3 105 20105 5 20 520 4 5 0

• The sequence of partial quotients is the same for these two pairs: 3, 5, and 4. This became a measure of ratio.

Measuring Ratios

35

67 21 421 4 14 1 04

335 105 201

3505 20 5

20 5 04

Continued Fractions67 3 21 421 5 4 14 4 1 0

67 4 13 3 2121 214

21 154 4

67 4 1 13 3 321 121 21 54 4

Continued Fractions

35

67 4 1 13 3 21 121 214 4

• There doesn’t seem to be any simple connection between the anthyphaireses of two ratios and those of their product or sum; thus writing ratios as sequences of partial quotients does not allow for any easy way to add or multiply them.

• The connection between the ordering of ratios and their anthyphairesis is more complicated than we might wish; it depends on the position of the first un‐matching partial quotient.

How Useful Is All This?

Anthyphairesis and Incommensurability

A

B C

D

E

F

G

No, Wait!

• You get to do that on you next Homework!!

• Prime, composite. • Every composite number is “measured” by some prime number.

• Primes are the basic “measures’ of other numbers.

• Many results of “elementary” number theory.• The infinitude of primes (IX – 20).

Number Theory, Books VII ‐ IX

The Infinitude of Primes

• You should really know this proof for the good of your soul.

• Circles are to one another as the squares on their diameters.

• Spheres are to one another in the triplicate ratio of their respective diameters.

• Any cone is a third part of a cylinder which as the same base with it and equal height.

• Notice: Not formulas.

“Areas” and “Volumes”

Regular Polyhedra

TetrahedronCube

Octahedron

Dodecahedron Icosahedron

The Elements

• Why are the Elements so important?• What legacy did they give us?

Documents

Greece: The Beginnings to Euclid - Mathematics Educationmathed.byu.edu/~williams/Classes/300F2011/PDFs/PPTs/Greece 1.pdf · particularly geometry, to Greece, from Egypt. ... of Geometry